Topic A: Exponential Functions & Sequences
Lessons 1–5 · Growth, decay, geometric sequences, and base e.
01–03 — Lessons 1–3
Exponential Growth & Decay
f(x) = a·bˣ models situations that multiply repeatedly.
- a: initial value (y-intercept, f(0) = a).
- b > 1: growth factor — increases exponentially.
- 0 < b < 1: decay factor — decreases toward zero.
- Continuous growth: f(t) = ae^(rt), base e ≈ 2.718.
- Doubling time: when f(t) = 2a → solve 2 = b^t.
Example: $1,000 at 5% annual growth: f(t) = 1000·(1.05)ᵗ. After 10 years: f(10) ≈ $1,629.
Practice — Lessons 1–3
Q1 · Multiple Choice
A population of 500 grows at 8% per year. Which function models this?
Q2 · Short Answer
Using f(t) = 200·(0.5)ᵗ, what is f(3)?
04–05 — Lessons 4–5
Geometric Sequences & Series
A geometric sequence multiplies by a constant ratio r each term.
- General term: aₙ = a₁ · r^(n−1).
- Common ratio: r = aₙ₊₁/aₙ.
- Finite sum: Sₙ = a₁(1 − rⁿ)/(1 − r), r ≠ 1.
- Infinite sum (|r|<1): S = a₁/(1 − r).
Sequence: 3, 6, 12, 24 … → a₁ = 3, r = 2 → a₅ = 3·2⁴ = 48
Practice — Lessons 4–5
Q3 · Multiple Choice
Find the 6th term of the geometric sequence 2, 6, 18, …
Q4 · Short Answer
For a geometric sequence with a₁ = 5 and r = 3, what is a₄?
Topic B: Logarithms
Lessons 6–10 · Definition, properties, and graphing logarithms.
06–07 — Lessons 6–7
Logarithms as Inverses of Exponentials
Logarithms undo exponentiation. log_b(x) = y means b^y = x.
- Common log: log(x) = log₁₀(x).
- Natural log: ln(x) = log_e(x).
- Inverse relationship: b^(log_b(x)) = x and log_b(b^x) = x.
- Domain of log: x > 0. Range: all reals.
log₂(8) = 3 because 2³ = 8. · ln(e⁵) = 5.
Practice — Lessons 6–7
Q5 · Multiple Choice
log₃(81) = ?
Q6 · Short Answer
Rewrite as a logarithm: 5² = 25. Write log_?(?) = ?
08–10 — Lessons 8–10
Properties of Logarithms
- Product: log_b(MN) = log_b(M) + log_b(N).
- Quotient: log_b(M/N) = log_b(M) − log_b(N).
- Power: log_b(Mⁿ) = n·log_b(M).
- Change of base: log_b(x) = log(x)/log(b).
- Use to expand or condense logarithmic expressions.
log(x²y/z) = 2log(x) + log(y) − log(z)
Practice — Lessons 8–10
Q7 · Multiple Choice
Expand: ln(x³/y). Which is correct?
Q8 · Short Answer
Use change of base to find log₄(64) exactly.
Topic C: Solving Exponential & Log Equations
Lessons 11–16 · Applying log properties to solve equations and model real situations.
11–14 — Lessons 11–14
Solving Exponential & Logarithmic Equations
Use logarithms to solve equations where the variable is in the exponent.
- To solve bˣ = c: take log of both sides → x = log(c)/log(b).
- To solve log_b(x) = c: rewrite as b^c = x.
- Check: for log equations, verify the argument > 0.
- Use ln when working with base e.
3ˣ = 20 → x = log(20)/log(3) = ln(20)/ln(3) ≈ 2.73
Practice — Lessons 11–14
Q9 · Multiple Choice
Solve 2ˣ = 32. What is x?
Q10 · Multiple Choice
Solve log₂(x) = 4. What is x?
15–16 — Lessons 15–16
Exponential Modeling in Context
Apply exponential and log functions to real-world growth, decay, and finance.
- Compound interest: A = P(1 + r/n)^(nt).
- Continuous compounding: A = Pe^(rt).
- Half-life: f(t) = a·(0.5)^(t/h), h = half-life period.
- Use logs to find when a quantity reaches a specific value.
Carbon-14 half-life ≈ 5,730 yr. After 11,460 yr (2 half-lives): 1/4 remains.
Practice — Lessons 15–16
Q11 · Multiple Choice
A substance decays to half its amount every 10 years. After 30 years, what fraction remains?
Q12 · Short Answer
$500 invested at 6% continuously compounded. Use A = 500e^(0.06t). After t = 0, A = ?